Topology of smooth and symplectic 4-manifolds

由于我们的世界包括时间在内都是四维的，通过几何和拓扑方法探索四维空间的异同可以更好地理解我们所生活的宇宙。四维空间上的光滑和辛结构在理论物理中具有广泛的特征；例如在经典力学中，弦和量子场论。该研究项目提出了新的想法和技术，为解决光滑和辛4流形的拓扑结构以及接触3流形及其填充的几个重要问题创造了有利条件。该课程的一个关键方面是将许多复杂的问题简化为曲面上曲线之间相当简单的代数关系，这为研究生和高级本科生提供了一个很好的基础。PI将参与并指导学生进行相关的研究课题。这是一个低维几何和拓扑学的项目，重点研究涉及辛4流形和接触3流形的各种深奥问题。一些关于4-流形的课题涉及辛Calabi-Yau曲面的分类、Lefschetz铅笔和纤维的多样性、奇异和辛4-流形的新构造以及各种稳定等价的复杂性。对于3-流形，PI将从开卷支撑格、新发现的具有任意大Stein填充的接触3-流形、3球中复杂曲线边界环的多样性以及拟正环的推广等方面探讨Giroux对应中的复杂性。规范理论、新的映射类群技术和辛手术将在本课程中发挥重要作用。

As our world, with time included, is four dimensional, exploring similarities and differences of 4-dimensional spaces through geometric and topological methods leads to better understanding of the universe we live in. Smooth and symplectic structures on 4-dimensional spaces are broadly featured in theoretical physics; e.g. in classical mechanics, string and quantum field theories. The research projects invoke new ideas and techniques, creating leverage to address several important problems regarding the topology of smooth and symplectic 4-manifolds, and that of contact 3-manifolds and their fillings. A key aspect of this program is the reduction of many intricate problems to fairly simple algebraic relations between curves on surfaces, which sets an excellent ground to present problems accessible to graduate and advanced undergraduate students. The PI will engage and mentor students in related research topics.This is a project in low dimensional geometry and topology, focusing on a variety of profound questions involving symplectic 4-manifolds and contact 3-manifolds. Some of the projects on 4-manifolds are pertinent to classification of symplectic Calabi-Yau surfaces, diversity of Lefschetz pencils and fibrations, novel constructions of small exotic and symplectic 4-manifolds, and complexity in various stable equivalences. As for 3-manifolds, the PI will probe the complexity in Giroux correspondence in terms of open book support genus, the characterization of newly discovered contact 3-manifolds with arbitrarily large Stein fillings, diversity of complex curves bounding links in the 3-sphere, and generalizations of quasi-positive links. Gauge theory, new mapping class group techniques and symplectic surgeries will play a vital role in this program.